Transducers employing mechanical elements, particularly microelectromechanical systems (MEMS) devices, are increasingly used in different applications to detect or control the movement of an object or the presence of a substance or condition due to their relatively small size and their capability to detect relatively small amounts or changes in the measured item. MEMS devices typically employ a movable, inertial mass or flexible membrane anchored to a rigid structure. For example, in a MEMS accelerometer, the inertial mass may be suspended in a plane above a substrate (or a portion of the substrate) and movable with respect to the substrate. In one common implementation, the substrate is a single crystal silicon wafer, and the mass is created through subsequent thin film deposition, patterning, and etch steps similar to those used in conventional integrated circuit processing. Motion of the inertial mass induced by applied force or acceleration may be sensed resistively, capacitively, optically, or by other transduction mechanisms. For example, commercial capacitive MEMS accelerometers that sense translational motion in the x-, y- and z-axes may have electrodes positioned above, below and/or on opposing sides of the inertial mass to allow measurement of differential capacitance in each axis.
Resonant frequency and quality factor are two important parameters used when characterizing a MEMS transducer, or other transducer employing a mechanical element. The resonant frequency of a system is the frequency or frequencies at which the system tends to oscillate or resonate at maximum amplitude in an ideal, undamped environment. The quality factor, or Q, is a dimensionless parameter that measures the effect of a resonant system's resistance to oscillation due to damping, with a higher Q system having a lower resistance. Thus, in MEMS sensors, the resonant frequency determines the sensitivity of the device while the quality factor dictates its likelihood of colliding with part of the rigid support structure in the event of a shock. Both parameters are required to predict the time-dependent response of the mechanical element to an arbitrary external stimulus.
MEMS devices are commonly made by a sequence of thin film depositions and etches performed on a silicon substrate, e.g., a single crystal silicon wafer or a silicon-on-insulator (“SOI”) wafer. Due to the nature of the MEMS fabrication processes typically used, many devices are generally overdamped in the z-axis (normal to the plane of the wafer) due to the strong squeezed-film damping effects. In overdamped systems (i.e., Q<0.5), the accurate measurement of these two parameters based on the frequency or transient response is usually difficult to make. One important consequence is that calculating the sensitivity based on electrical measurements cannot be done accurately. For MEMS accelerometers, a common alternative is to submit each MEMS device to a shake test in order to determine its sensitivity to acceleration, which entails substantial characterization costs. As known by those skilled in the art, the shake test entails physically shaking or vibrating every device at one or more predetermined frequencies and amplitudes and then measuring its voltage output. The device is then appropriately calibrated so that each sensor has approximately the same sensitivity regardless of variations in the physical components that make up the sensor.
Characterization of an underdamped second-order system (i.e., Q>0.5) is typically accomplished using a frequency sweep and measuring the peak frequency and width. This is possible because the poles in an underdamped second-order system are imaginary, resulting in a peak of the frequency response near the resonant frequency. However, for an overdamped second-order system, the two poles are real and at different frequencies. As known by those skilled in the art, poles represent frequencies where the response undergoes an abrupt transition. In MEMS transducers, it is not uncommon for the poles to span an order of magnitude of frequency or more. Thus, historically, the location of both poles needs to be measured in order to calculate the resonant frequency and Q factor of an overdamped system based on the frequency response. The measurement of the higher frequency pole, however, is difficult and costly to determine since it requires substantial dynamic range and sufficiently high bandwidth measurement circuitry.